# What is the speed when a ball hits the ground

1. Jan 19, 2012

### chris_0101

1. The problem statement, all variables and given/known data

A sphere of a mass m and radius R is fired straight up with a speed v0
a) What is its speed when it hits the ground (Include quadratic air resistance but ignore linear air resistance

b) Evaluate the result for part (a) with R = 0.2c, v0 = 1.3m/s and m = 100mg

I am puzzled with regards to whether or not I should include the upward motion and solving the question entirely

2. Relevant equations

mg -cv2 = m$\frac{dv}{dt}$

3. The attempt at a solution

After following my class notes, I found that the final velocity as a function of time is:
v(t) = vterminal*tanh(gt/vterminal})

Any help is greatly appreciated,

thanks

2. Jan 19, 2012

### Simon Bridge

How could you determine the extent to which the upwards part of the motion is important? What would this determine? Do you need it?

Note: air resistance will increase acceleration when going up and decrease acceleration going down, so whatever you do you'll want to break the motion into two parts.

3. Jan 19, 2012

### chris_0101

Wait, how can air resistance increase acceleration when going up. From what I have understood, air resistance slows down the object flying upward, likewise with gravity.

I do understand that I have to break up the question into the upward and downward parts, but the question asks for the speed just before it hits the ground. Furthermore, the object being shot with an initial velocity will eventually go to rest at its highest point, then fall back down again. This is what caused me to believe that the upward component is negligible.

4. Jan 19, 2012

### Simon Bridge

acceleration is a vector.

Without air resistance, the only force is gravity (weight), pointing down.
The acceleration vector, therefore, points downwards.

When the ball goes up, the force of the air resistance points down, adding to the force of gravity and so increasing the acceleration.

When the ball goes down, the air resistance points up, subtracting from the force of gravity and so reducing the acceleration.

When the ball falls from it's highest point to the ground, what determines the speed when it strikes?
Perhaps the ball has enough time to reach it's terminal velocity? How would you know?
Don't you need to know how high it started from?